Approximate spectral absorption coefficients for pure rotational transitions in diatomic molecules

J. Quant. Specfrosc. Radfat. Transfer. Vol. 2, pp. 201-214.

Perpamon Press Ltd. Printed in Great Britain

APPROXIMATE SPECTRAL ABSORPTION COEFFICIENTS FOR PURE ROTATIONAL TRANSITIONS IN DIATOMIC MOLECULES S. A. GOLDEN Rocketdyne, a Division of North American Aviation, Inc. 6633 Canoga Avenue, Canoga Park, California (Received 19 December 1961) Abstract-Approximate spectral absorption coefficients for the pure rotational spectra of diatomic molecules have been computed by using (a) the “just-overlapping line” model, and (b) a model involving smearing of the rotational structure. As has been shown previously for the infrared vibration-rotation spectra and for the electronic band spectra, the two models yield effectively identical results. Numerical data are presented for OH, CO, NO, HCl, HBr, and HF. I. INTRODUCTION

THE theoretical calculation of spectral absorption coefficients and of spectral and total emissivities has been discussed in the recent literature.(r-6) Perhaps the simplest models, that appear to yield acceptable approximations to the spectral absorption coefficients of diatomic gases at elevated temperatures, are the “just-overlapping line” model or a model using a smeared-out rotational fine structure.{ 397) Reference to the published literature shows that these two models yield identical results both for infrared vibration and for electronic band systems. This conclusion for the vibration-rotation bands may be demonstrated by comparing the formulas given in section 11-S of reference 1 with the analysis of reference 3 and for the electronic band systems case by comparing equation (14-34) of reference 1 with equations (5) and (6) of reference 6 (see also reference 7). We shall show that these two models also yield identical results for the pure rotational spectra when they are used to the same degree of approximation. The theoretical results for the diatomic molecules presented in the following sections constitute progressively better approximations as the temperatures and pressures are raised. Similarly, for a given temperature and pressure, the assumed models and the calculated results will constitute better approximations for those diatomic molecules with relatively close rotational line spacings. II.

OUTLINE

OF THEORY

The model with smeared-out rotational structure is discussed to the rigid-rotator approximation in section A; the “just-overlapping line” model is described to the same approximation in section B. In section C, vibration-rotation effects are demonstrated for the extreme case of the HF molecule by replacing Be with Bv in the smeared-line model and also by using Herman-Wallis matrix elements in the “just-overlapping line” calculation. E

201

202

S. A. GOLDEN

A. The smeared-line model In the rigid-rotator approximation for a diatomic molecule, a pure rotational transition from the v-th vibrational level, (v, J) -+ (v, J- 1) is characterized by the wavenumber w = 2JBe. The rotational

part of the total energy of the (v, J- I)-state is EJ-1 = J(J-1)hcBe

= hcw(w-2B,)/4Be.

(1)

In the smeared-line model,@1 59 69 7) the contribution to the total f-number v --f v transitions, in the wavenumber interval between w and w + dw, is

Wv,2,

for all

-fv,$$xP - (EJ-dk T)]

-=

dw or, using equation (l), dfv, v -

do

hc(w - Be)

EJ-1

=fv,v 2BekT =

f

exp (---> kT

hc( w - Be) *”

exp(s)

exp-

[-z&F].

(2)

2BekT

The total rotational f-number for the v-th vibrational moment matrix element of the transition by the relation

level is related to the dipole

876 mc

f&V= ~~2+w12; but, for a pure rotational

transition in the rigid-rotator

model,

w-Be

= 2(wwhere ~0 is the permanent written as dfv, -=-- v

dw

dipole moment of the molecule. Thus, equation (2) may be

27Grn~2~~~(~-2B~) BekT 3 e2

The spectral absorption

Be)w2

exp($$

exp[-

hCE-$2].

coefficient associated with a particular

VlJv, v

re2 NV dfv, = m>p

(3)

v + v transition is

v

-[l-exp(-g)] dw

where NV/p is the number of radiators in the v-th vibrational state per unit volume per unit pressure. Substituting equation (3), the spectral absorption coefficient is found to be 2rr3 NV /L+I~w(w-~B~) hcBe (Po)v, v = 3-y exp (-)4kT Bek T x

exp - hcLit)2] [

[I-exp(

-g)].

(4)

Approximate spectral absorption coefficients for pure rotational transitions in diatomic molecules

203

Equation (4) yields the spectral absorption coefficient for a particular vibrational state; in order to find the total spectral absorption coefficient, (P&, Vmust be summed over all vibrational states. But, to the order of approximation used, the only term in the expression for (P,)V, o that depends on v is NV/p. Therefore, the summation yields simply the total number of radiators (in all states) per unit volume per unit pressure, N/p. Thus the total spectral absorption coefficient becomes P,

=

2773;j$w( w - 2BJJ --(kT)2

exp (z)exp-

[ hc~~~)“]

3Be X [1-exI+)]

(5)

where the ideal gas law has been assumed so that N/p = l/kT. B. The “just-overlapping line” model In the “just-overlapping line” model,(ls 2) the spectral absorption coefficient of a line corresponding to the transition (v, J) --f (v, J- 1) is replaced by the local average value of the spectral absorption coefficient, viz., G,J), (v,J-1) <

P@,J),(V,J-1) >

=

(Pdv, 2,

=

6

where SC,,J), cV,~-1) is the integrated absorption of the line and 6 is the local average of the rotational line spacing. In the rigid-rotator model, the line spacing is 2B, and the integrated intensity is given by the expression 49 NV /L+~w(w-~B~) $v,

J), (v, J-1)

=

3

kT

P

x [I-exp-(g)].

Hence

279 NV /.Lo~w(w-~B~) <

p(v,

J),(v,J-1)

>

=

which is identical with equation (4) when equation (1) is used for EJ. We have thus shown that the smeared-line model gives results which are identical with those derived from the “just-overlapping line” model for the pure rotational spectrum when the rigid-rotator model is used in both cases. C. Efict

of vibration-rotation interactions

A &t-order correction to the expression for P, may be obtained by considering the variation of the rotational constant with v. This is done by replacing Be by Bv = Be-a,(~+$)

= Be

ae(v + 4) B I e

(6)

204

S. A. GOLDEN

in equation (4). The final summation over (P&, e, may be performed by retaining, only fist-order terms in ae(v + 3)/& after the exponentials and l/B0 are expanded. It is found that the resulting corrections are not large although they account for most of the difference between values of P, derived from equation (5) and from more exact calculations which are discussed below. A more exact calculation, based on the “just-overlapping line” model, has been performed for the HF molecule by using the following procedure: (a) We first evaluate the integrated intensities of the rotational lines in various vibrational levels by generalizing the approach of HERMAN and WALLIS@). Details describing these calculations are given in the Appendix. (b) We next choose a wavenumber interval centered on a strong rotational line by the method illustrated in Fig. 1. The integrated intensities of all lines lying in the specified interval are then summed to obtain a “total” integrated intensity over the interval.

FIG. 1. Illustration of the method of determining the local line spacing. Positions of the centers of very intense lines are designated by WI, wa, US, and W. The spacing used for the interval about the line whose center is at WI is AW = t(W- ~~-1)+3(wa+l--Wi) =

t(w+l-OC-l)

(c) The spectral absorption coefficients are finally set equal to the ratios of the sums of the line intensities to the assumed widths of the intervals, i.e., P, =

,,,cnSt w

I

Aw

These more exact calculations have been carried out only for HF. This molecule was chosen since the errors resulting from the use of the simplified formulas, in this case, are expected to be larger than for any of the other molecules considered. Spectral absorption coefficients for HF have also been computed by use of equation (5). Fig. 2 compares the results of the smeared-line model calculations with those of the “just-overlapping

Approximate

spectral absorption

coefficients for pure rotational

transitions

in diatomic molecules

205

line” model with essentially exact matrix elements and rotational energies. Reference to Fig. 2 shows that the simple relation, given in equation (5), constitutes an excellent approximation to the more exact numerical calculations even for the extreme case of HF. 50

2.0

---JUST

OVERLAPPING

LINE MODEL HF

I-0

TEMPERATURES T

05

/-

(‘K)

5oc

5 k T’3

0.2

g

@’

; LL

0.05

3 z

0.02

g

0.01

5: $

0.005

d F

0.002

g 03 0.001

O~OOO?

O~OOOi

0.000

I

I

I

I

200

400

600

WAVE

II

NUMBER,

I 600

\

I 1000

\\I\

\ 1200

I

CM-l

Fro. 2. Comparison of the results of the smeared-line and just-overlapping-line models. The smeared-line calculation is done to the rigid-rotator approximation while the justoverlapping-line calculation utilizes essentially exact line intensities and rotational energies.

The errors resulting from the use of equation (5) became significant only at relatively high temperatures and/or large wavenumbers. This is precisely as expected since the major contributions to P, at high temperatures and large wavenumbers are from states with large J-values; it is exactly these states for which the simple rigid-rotator model will not be valid.

III.

RESULTS

The results of calculations of P,, based on the use of equation (5) for OH, CO, NO, HCI, and HBr are summarized in Figs. 3 to 7, respectively. The numerical data used in the calculations are shown in Table 1.

S.

206

A. GOLDEN

TABLE1. VALUESOF THEROTATIONAL CONSTANTS AND PERMANENT DIPOLE MOMENTS USEDIN THE SMEARED-LINE CALCULATIONS

B,(O) OH co NO HCl HBr HF

CL0 --

18.871 I.931 1.705 10.591 8.473 20.939

154(b) 0.118(c) 0.148(d) 1*085(@ 0.82(r) 1*736(E)

(a) All values of the rotational constants were taken from G. HERZBERG,

(b) (c) (d) (e) (f) (g)

Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules (2nd edition), Van Nostrand, New York (1950). R. P. MADDEN and W. S. BENEDICT,J. Chem. Phys. 23,408 (1955). B. J. RANSIL,J. Chem. Phys. 30, 1113 (1959). A. A. MARYOTTand S. J. KRYDER,J. Chem. Phys. 31, 617 (1959). R. P. BELL and I. E. Coop, Trans. Faraday Sot. 34, 1209 (1938). B. SCHIJRINand R. ROLLEFSON, J. Chem. Phys. 26, 1089 (1957). G. A. KUIPERS,J. Molec. Spectroscopy 2, 75 (1959).

In view of the noted acceptable agreement with more exact calculations for HF (Fig. 2), we would expect the results shown in Figs. 3 to 7 to constitute reasonably good 5:o

2:o I.0 a5 5 2$

0.2

ii -

0.1

i

0:05

t 8 z 9 t 8 9

a02 0.0 I oao5

d -$

0002

L v)

0001 0~0005

00002 @OOOl

I 200

I

400

I\

600

WAVE NUMBER;

I \

800

I\

looo

CM-*

FIG. 3. Results of the smeared-line calculations for OH.

150

NUMBER,

100

WAVE

50

I

I

I CM-’

200

\ I 250

I\ 300

2500

2000

1500

0Oc.

500

co

3s 3

\I

TEMPERATURESCK)

FIG. 4. Results of the smeared-line calculations for CO.

0~0001

0.0 002

0.0005

0.001

0002

0005

I

10-5

-

50

n

I50

WAVE NUMBER.

IO0

CM-’

200

250

300

FIG. 5. Results of the smeared-line calculations for NO.

I x 10-G

2x104

5 x 10-G

I x 10-S

2xiO-5

5 x

0~0001

0~0002

0.0005

0.00

DO02

0.005

0.01

350

-

~.

a

B 2 R

g

c

8

a

E 0

208

S.A.GOLDEN

,_WlV,_W3

E

‘lNXMd303

NOlldMOSBW

lWU33dS

Approximate spectral absorption coefficients for pure rotational transitions in diatomic

molecules 209

approximations to the spectral absorption coefficients provided that the temperatures and pressures are high enough to make the spectral line width at least comparable with the ratio of the wavenumber interval to the number of relatively strong contributing lines. This condition is probably not satisfied for any of the hydrogen halides or OH at atmospheric pressure and temperatures of the order of 2000°K. However, it should be reasonably well satisfied for CO and NO under the same conditions. For isolated strong lines, such as exhibited by the hydrogen halides at atmospheric pressure, a non-overlapping line model should be more realistic than the models discussed in the present paper. Acknowledgements-The author wishes to acknowledge the continued encouragement and stimulation of Rocketdyne personnel, especially of Messrs. J. E. WITHERSP~~Nand F. S. SIMMONS. REFERENCES 1. S. S. PENNER, Quantitive Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading Mass. (1959). 2. V. R. STULL and G. N. PLASS,J. Opt. Sot. Amer. 50, 1279 (1960). 3. S. S. PENNER,K. SULZMANNand C. B. LUDWIG,J. Quanf. Spectrosc. Radiat. Transfer 1, 96 (1961). 4. W. MALKMUSand A. THOMSON, J. Quant. Spectrosc. Radiat. Transfer 2, 17 (1962). 5. B. KIVEL, H. MAYERand H. BETHE,Ann. Phys. 2,57 (1957). 6. J. C. KECK, J. C. CAMM,B. Krvs~ and T. WENTINK,Jr., Ann. Phys. 7, 1 (1959). 7. R. PATCH,W. L. SHACKLEFORD and S. S. PENNER,Technical Report No. 4, Contract AF49(638)-984, California Institute of Technology, Pasadena (1961). 8. R. HERMANand R. F. WALLIS:J. Chem. Phys. 23, 637 (1958). 9. F. B. HILDEBRAND,Introduction to Numerical Analysis, McGraw-Hill, New York (1956).

APPENDIX CALCULATION

The vibration-rotation

OF THE EXACT MATRIX AND LINE STRENGTHS

ELEMENTS

matrix elements,

(A-1) where &(TJ) is the vibration-rotation wavefunction of the (v, J)-state and M(r) is the dipole moment operator, have been calculated by a method similar to that described by HERMAN and WALLIS.@) They give explicit expressions for the wavefunctions of the v = 0, 1, and 2 states and also explicit expressions for the matrix elements of the 0 & 1, 0 -+ 2, and 1 + 2 vibrational transitions. Rather than use those expressions for all transitions, or to assume the harmonic-oscillator model to compute the higher order matrix elements, as has been done by STULL and PLASS(~) and by MALKMUS and T~MPsoN(~), we have used equation (A-l) directly. The first step in the use of equation (A-l) requires determination of the general wavefunction S,(~J). Using standard perturbation theory and the notation of HERMAN and WALLIS@), the wavefunctions

are given

by

SV,J = SV,JO+

$ m-0

me

C~,,~,JS~,JO

S. A. GOLDEN

210

where m s S,, J~H’&,J~~~

+J

C v.m,J

s Ev,~O-Ev,~’

Sv, JO is the solution rotator model), H’ is potential energy, and (v, J)-state. Retaining

’

to the unperturbed system (i.e., the harmonic-oscillator-rigidthe perturbation operator corresponding to the cubic terms of the J(J+ l)/ r2 expansions, and E,, Jo is the unperturbed energy of the terms up to ys = 4Be2/we2, the wavefunctions are

J

(v+1)(v+2)(v+3)bJ,s~3

S v,J = &Jo-$

- 2 d(V

Jo

2 + 1)(V+2)bJJ(J+

l)~‘2&+2,Jo

(A-2) +

$w)bJJ(J+

1)~~‘~&-2,

l)(v-2)

v(v-

+Q

Jo

J

2

b'&-s,J".

Because of the approximations made in the perturbation calculations, is not quite normalized. In order to have normalized functions, SV,J the constant ii?,J

=

equation A-2 be divided

l+~J252(5+1)2y3(l+v+v2) [

1 I2

+ ;bJ’2(87+256V+2k’2+

164~2)

. I

The values of NV,J for small v and J are very close to unity (Table A-l) and so can be neglected. However, for large v, the difference from unity becomes quite important and serious errors can result if the normalization constant is ignored. TABLE&~.

J/v 0 5 10 15 20 25 30 35 50

VALUES OFTHE

0 1mo10 1.00008 1XQOO6 1mOO3 1GOOOO 1aOOO1 1*00010 1+0040 la0122

N~~~ALI~A~ONCONSTANT

4 1.01694 l-01496 1.01028 1a0454 1mO41 1a0171 1.01370 l-04368 1.10178

N”,J FOR HF

8

12 -

16 -__-

1.10630

1.30683 1.27450 1.19492 l-08977 1.00812 1.03482 1.30507 1.79052 2.46170

1.61998 1.55981 140772 1.19591 1.01841 1.07823 1.50975 2.23640 3.18414

la9431 l-06553 1.02929 1GO261 1.01115 1.08531 l-25035 1.51984

Approximate

spectral absorption

Using the wavefunctions operator expansion,

coefficients for pure rotational

transitions

in diatomic molecules

given in equation (A-2) and the conventional

211

dipole moment

equation (A-l) may be written in the form

(A-3) where f(u) is a polynomial of order (v + v’ +i+ 6). Many attempts were made to find a general expression for the integrals appearing in equation (A-3) but all such attempts were uniformly unsuccessful. In actual practice, the integrals are evaluated numerically by an M-point Hermite-Gauss quadrature@) where M 2 &(v+ v’ + i + 7). As HILDEBRAND@), shows, this criterion for the number of points used in the quadrature insures that the numerical evaluation will be exact (except for round-off errors). The results of this calculation (including computed values of the integrated absorptions) are given in Tables A-2, A-3, and A-4 for pure rotational transitions with v = 0, 1, and 2, respectively for the HF molecule. Needless to say, these calculations were performed using an electronic computer. The programs which were developed to do the calculations were written in a general form and will perform the equivalent calculation for an arbitrary transition (v, J) -+ (v’, J+ 1). To date this calculation has been done only for HF with v’ - v = 0, 1,2,3 and 4; the other molecules discussed above will be similarly treated in the immediate future, and the results will be presented in a forthcoming publication. Copies of the more detailed tabulated results and copies of the FORTRAN programs can be made available to any interested party.

;: 37 38

::

;: 25 26 21 28 29 30 31 32

:: 17 18 19 20 21 22

: 5 6 I 8 9 10 11 12 13 14

0 1 2

_----------

J 4.8689E 00 3.2653E 01 8*2157E 01 le2915E 02 1.4889E 02 1.3525E 02 1 TIO62E 02 6.2767E 01 3.3349E 01 1.5261E 01 6.0652E 00 2.1075E 00 6.4383E-01 1*7374E-O1 4.1588E-02 8 ~8649EL-03 1 e6888E-03 2*8852%04 4.4353E-05 6-l 549E-06 7.735OE-07 8.8315E-08 9*1902E+O9 8.7442E-10 7.6317E-11 6.1297E-12 4.5455E-13 3.1225E14 1.9936515 1*1871E-16 6.6143E-18 3*4609E-19 1.706&20 7*9583E-22 3.5226E-23 1.4854E-24 5*9898E-26 2.3184E-27 86472E-29 3.1202E-30 0.0

1.7334E 00 1.7338E 00 1.7346E 00 1.7356E 00 1.737OE 00 1.7387E 00 1.7406E 00 1,7428E 00 1.1453E 00 1.748OE 00 1*75llE 00 1*7543E 00 1.7578E 00 1.7615E 00 1.7655E 00 1*7696E 00 1.774OE 00 1*7785E 00 1*7832E 00 1*788OE 00 1.793OE 00 1.7981E 00 1.8034E 00 1.8087E 00 1.8142E 00 1.8198E 00 1.8254E 00 1.8311E 00 1.8368E 00 1.8426E 00 1.8485E 00 1.8544E 00 1.8603E 00 1.8662E 00 1.8721E 00 1.878OE 00 1.884OE 00 1.8899E 00 1.8958E 00 1*9017E 00 1.9075E 00

T = 2000°K 7.49693-02 5.7338Erol 1*7956E 00 3.8332E 00 6*5454E 00 96005E 00 1.2566E 01 1.5017E 01 1*6631E 01 1.7245E 01 1*6867E 01 1.5649E 01 1*3832E 01 1.169OE 01 9*4727E 00 7.3784E 00 5.53618 00 4.0086E 00 2.806OE 00 19016E 00 12494E 00 7.9698EOl 4.9416E-01 2.9819E-01 1*7531E-O1 1 GO53E-01 5.629633-02 3.0815ELO2 1.6505E-02 8.659-3 4.4549EO3 22495E-03 1.1161E-03 5 44593-04 2.6162EW4 1.2386JGO4 5.7849E-05 2.668OE-05 1.2164E-05 5.4873E-06 2.4521E-06

T = 1500°K 1.8422E-01 1*3884E 00 4.2428E 00 8*7528E 00 1.4303E 01 1*9885E 01 24434E 01 2.7153E 01 2.7701E 01 2.6213E 01 2.3184E 01 1.9273E 01 1*5129E 01 1.1255E 01 7.9587E 00 5.3638E 00 3*4533B 00 2.1281E 00 1.2577E 00 7~1409E-01 3*9OllE-O1 2.0538&01 1 Q435E-01 5.1244E-02 2.4355E-02 1.1218E-02 5.0141&03 2.1778E-03 9.203oE-M 3.7889E-04 1.5217FLO4 5.9691EM5 2.29OOFL-05 86038E-06 3.1696%-06 1.1465EO6 4.0769E-07 1 a4272E-07 4.925OE-08 1 a6776E-08 56483E-09

at the temperatures

2*8591E 00 2.0988E 00 1 *SO24E 00 lQ498E 00 7*1692FGOl 4.7893E-01 3.1328E-01 2*0085E&Ol 1.2632E-01 7 *8008E--O2 4.7341&02 2*8258EO2 1 a6605EC-02 9.6131E-03 5.4879Ero3 3Q918E-03 1 e7205E-03 9*4637%04 5.1502EX4 2.7752E-04 1*4819E-W 7.8485EM5

3647OE-02 2.8141E-01 8.943OE-01 1*9488E 00 3.4167E 00 5*1754E 00 7.0361E 00 8*7839E 00 1.022OE 01 1*1196E 01 1.1633333 01 1*1528E 01 1.0943E 01 9.98443 00 8.7806E 00 746MiE 00 6.1368E 00 4.89555 00 3.79343 00

T = 25OO“K

J. Gem. Phys. 34,420 (1961).

Values of the dipole moment expansion coefficients were taken from the work of G. A. KUIPERS, J. Molec. Spectroscopy 2, 75 (1958).

JR., J. J. BALL and N. ACQUISTA,

6.2841E-01 4~5992E QO 1*3385E 01 2.579OE 01 3*8608E 01 4.8232E 01 5*224lE 01 5.02238 01 4*3493E 01 3*4293E 01 2.4811E 01 1.6569E 01 1.0263E 01 5.9188E 00 3*1891E 00 1.6101E 00 7.6368E-01 3~4111EYOl 1 a438OE-01 5.7341E-02 2.167OE-02 7.7763Ea3 2.65.5OE-03 86405EL-04 2.6853E-04 7.984oE-05 2.2752E-05 6.2254EO6 1.6386E-06 4.1563EM7 1.0178E-07 2.4109Ero8 5.5336E-09 1.2331E-09 2.6726%10 5645OE-11 1.1642%11 2.3488&12 4%452E-13 9*023OE-14 1.725OE-14

T = 1000°K

Line intensities (in atm-km-9

constants were taken from the work of D. E. MANN, B. A. THRUSH, D. R. LIDE,

T = 5OO“K

(in Debyes)

Mat. elt.

POSITIONS, MATRIX ELEMENTS, AND LINE INTENSITIES OF LINES IN THE R-BRANCH OF THE VIBRATIONAL TRANSITION I’ = 0. TO Y = 0 IN HF-19

Values of the positional spectroscopic

41.113 82.174 123.134 163.942 204.541 244900 284.952 324.654 363.958 402.819 441.188 479.024 516.280 552.916 588.889 624.180 658.692 692444 125.384 157.476 788.686 818.985 848.341 816.123 904.107 930.464 955.768 919.995 1003.118 1025.117 1045 ~966 1065642 1084.121 1101.381 1117.395 1132.139 1145.584 1157.705 1168467 1177.840 1185.786

Position (in l/cm)

TABLE A-2.

-

-

B 6 B

?

.P

h) ;3

z;

:: 36

::

;; 28 29 30 31

;: 25

;

;:

:; 18

:: 15

:; 12

z 7 8 9

:

0 1 2

J ----

-

Mat elt. (in Debyes) --1*7278E 00 1.7283E 00 1*7291B 00 1.7302E 00 1.73163 00 1.7334E 00 1/7354E 00 1.7378E 00 1.74043 00 1*7433E 00 1.7465E 00 1.7499E 00 1.7536E 00 1*7576E 00 1*7618E 00 l-76623 00 1*7708E 00 1*7756E 00 1*7806E 00 la7858E 00 1*7912E 00 1.7967E 00 1.8024E 00 1*8082E 00 1*8141E 00 1.8201E 00 l-82633 00 1*8325E 00 1*8389E 00 1.8453E 00 1*8518E 00 1*8583E 00 1.8649E 00 1.8716E 00 1*8783E 00 1*8851E 00 1.8919E 00 1*8987E 00 1.9056E 00 1*9125E 00 1*9194E 00

0.0

5.0279E-05 3.39423-04 8 a634OE-04 1*3782Ea3 1.6203E-03 1 m5075E-03 1.1537E-03 7.4351E-04 4.0986EW4 1.9542EO4 8.1262E-05 2.9668JW5 9.5621E-06 2.7335E-06 6.9596EO7 1*5843EO7 3.2359E-08 5.9503E-09 9.8831510 1*4875ElO 2*035OE-11 2*5387E-12 2.89703-13 3*0334E14 2.9237E15 26021E16 2.1455E17 1644OE18 1*1747E-19 7.8529E-21 4*9278E-22 2.9128E23 1.6273E24 8.6238E26 4.3503E27 2Q967E28 9.6899E-30 4*3105E-31 0.0 0.0

--____~ T = 5OO’K --1*9372E4)3 1.4226E-02 4.163OE.02 8.0836E-02 12222E-01 1.5455E-01 1 a6983E-01 1.6596E-01 146423-01 1*1787EOl 8.7252E-02 5.9743%02 3.8019E-02 2.2575E-02 1.2548E-02 6.549OE-03 3*2174E+O3 1.4914E-03 6.5379E-04 2.7159EXl4 1*0712E-C4 4.0198E-05 1*4377EO5 4.91OOE-06 16041E-06 5 eO224E-07 1.5097E-07 4.3644E-08 1.2157E-08 3.2683E-09 8*4965E10 2.1397E10 5.2293Ell 1*2426E-11 2*8759E-12 6*4959E13 1*4346E-13 3*1037E-14 6*591OE15 1.37653-15 2.8333E16

T = 1000°K -3.993OE-03 3.0592E-02 96069E-02 2.0589E-01 3.5333E-01 5.2143E-01 6.8744E-01 8*2844X%01 9.261OE-01 9.7039E-01 96014E-01 9.0209E-01 8.0834E-01 6.9324E-01 5.7066E-01 4*52OOEOl 3.452OE-01 2.5468E-01 1.8182E-01 1 a2578E-01 84449E-02 5.5094EO2 3.4969E-02 2.162OE-02 1 a3035E-02 7.67233-03 4.4133E-03 204835EO3 1.3687EO3 7.3944E-04 3.92OlE-04 2.0414E-04 1 eO453E-04 5.268OE-05 2.6157E-05 1.2808E-05 6.1911E-06 2.9572E-06 1 e3972E-05 6.5367E-07 3.0311E-07

T = 2000°K

at the temperatures

3.7955E-03 2.867OE-02 8.7936E-02 1.8235E-01 29999E-01 4.2045E-01 5.2161E-01 5.8609UIl 6.0543EOl 5.8094EOl 5.2174E-01 4~4106EOl 3.5256E-01 2.6744EOl 1*9311E-O1 1.3307E-01 8.7716E-02 5.5417E-02 3.3619E-02 1.9618E-02 1.1029E-02 5*9827EO3 3.1359E-03 1*5905E-O3 7.8162E-04 3*7268EO4 1 e7264E-04 7.7793E-05 3*4144EO5 1.4615E-05 6*109OE-O6 2.4967%06 9.9892E-07 3.9177E-07 1.508OE-07 5.7048&08 2.1236E-08 7*7893Ero9 . ^^ 2.8188E;uv 1.0078W 3*5646E-10

T = 15WK

Line intensities (in atm-‘cm-x)

POSITIONS, MATRIXELEMENTS, AND LINEINTENSITIES OF LINESIN THER-BRANCH OF THEVIBRATIONAL TRANSITION Y = 1, TO Y = 1 IN HF-19

3.4344E-03 2.6537E-02 8.4523E-02 1.8477E-01 3.2524Eal 4.9508E-01 6.7698E-01 8.5081E-01 9*974OE--Ol 1.1018E 00 1*1555E 00 1.1568E 00 1*1102E 00 1.02493 00 9.128OE-01 7.8605E-01 6.5585E-01 5*3111E-O1 4.1809E-01 3*2038Eol 2.3929E-01 1~7441E-01 1.2419E-01 86475E-02 5 -8947E-02 3.9372E-02 2.5792E-02 1.6586E-02 1 eO479E-02 6.51073-03 3.9811Ea3 2.39793-03 1.4238E-03 8.3412E-04 4.8253E-04’ 2.7586E-04 1.5598333-04 8.7303E-05 4.8409E-05 2.6613E-05 1.4519E-05

T = 2500°K

---____-

Values of the positional spectroscopic constants were taken from the work of D. E. MANN, B. A. THRUSH,D. R. LIDE, JR., J. J. BALL and N. ACQV~STA,J. Chem. Phys. 34,420 (1961). - __Values ,.__ -. of the dipole moment expansion coefficients were taken from the work of G. A. KUIPERS,J. Molec. Spectroscopy 2, 75 (lY58).

39.566 79.083 118.501 157.771 196.844 235.672 274.206 312401 350208 387.583 424.482 460.859 496.674 531.884 566,449 600.330 633.491 665.892 697.502 728.285 758.208 787.241 815.353 842.516 868.703 893.884 918.036 941.133 963.148 984.061 1003.843 1022.474 1039.927 1056.177 1071200 1084.967 1097.451 1108*620 1118443 1126.883 1133905

Position (in l/cm)

TABLEA-3.

-

3’: 36 37 38 39 40

14 28 29 30 31 32 33

%!

z; 23

:; 14 15 16 17 18 19 20

; 10 11

i 5 6 7

:.

0

J -----

---

1a72233 00 1.7228E 00 1.7237E 00 1.7248E 00 1*7263E 00 1.7281E 00 1.7303E 00 1.7327E 00 1.735533 00 1*7385E 00 1.74193 00 1.7455E 00 1.7494E 00 1.7536E 00 1*758OE 00 1*7626E 00 1.7675E 00 1.7726E 00 1.7779E 00 1.7834E 00 1*7891E 00 1*795OE 00 1.8011E 00 1.8073E 00 1.813jE 00 1.82023 00 1.8269E 00 1*8337E 00 1.8406E 00 1*8477E 00 1.8548E 00 1.8621E 00 la8695E 00 1*8771E 00 le8847E 00 1*8925E 00 1.9004E 00 1.9084E 00 1*9165E 00 1.9247E 00 1*9331E 00

Mat. elt. (in Debyes) --

T = 1000°K 7.6441506 5.6317EO5 1.6569EX4 3.2416E-04 4.9489E-04 6.3321E-04 7.0555E-04 7.0062Ero4 6.2944E-04 5.1705JW4 3.9136E-04 2.7456E-04 1.7939E-04 1.0958EO4 6.2788E-05 3.3845EO5 1.72OW5 8.2698E-06 3.7657E-06 1 a628OE-06 6.6952E-07 2*6242EX7 9.8208E-08 3.5157E-08 1~2061E-08 3.9717E-09 1.2578JZ-09 3*8373E-10 1*1298E-10 3.2156Ell 8.8641E12 2.3706%12 6.1621E13 1.5596%13 3.8508E14 9.2922E-15 2*1955E-15 5.0890&16 1*1594E-16 26017E-17 5*7612E-18

T = 500°K 8*514OE-10 5.7846E-09 14872E-08 24094E-08 2.8875E-08 2.7498E-08 2.1631E-08 1.43903-08 8.2221E-09 4.08OlE-09 1.7732E-09 6.7931E-10 2*3067E-10 6.9752E-11 1.886OE-11 4.5773E-12 l W%E-12 1.9768E-13 3.5408E14 5.7684E15 8.5738E16 1.1662E16 1+4562E-17 1.6743E-18 1.7781E-19 1.7497E-20 1%003E-21 1*3648E-22 1.0889%23 8*1535%25 5.7491E26 3*83OlE-27 2.4191E-28 1*4537E-29 8*3399%31 0.0 0.0 0.0 0.0 0.0 0.0 9.2162E-05 6.9769EO4 2.1477EHI3 4.4761E-03 7.4114E-03 1 W7OE-02 1.3111E-02 1~48908-02 1.5569E-02 1.5143E-02 1.3804E-02 1.1861E-02 9.6501E-03 7.4605Eco3 5.4975E-03 3.8712E-03 2.6109E-03 1.6899E-03 1.0517E-03 6.3029E-04 3.6438E-04 2.035OE-04 1.0995E+I4 5.7551E-05 2.9222E-05 14412E-05 6.9132E-06 3.2294E-06 1.4709E-06 6.5412E-07 2.8434Eco7 1.2098E-07 5.0439E-08 2~0635E-08 8.294OEO9 3.2793E-09 1.2771E-09 4.9054E-10 1.8607E-10 6.9792E-11 2*5922E-11

2.4052E-04 1.8458E-03 5.812OESO3 1.2503E-02 2.156233-02 3.2OlOEa2 4.2498E-02 5.1628E-02 5 e8246E-02 6.1656E-02 6.1693E-02 5.8678E-02 5.32823-02 4.6353Ea2 3.8744E-02 3*1191E-O2 2.4236E-02 1 a8209EC-02 1.3251E-02 9.3531E-03 6.41293-03 4.2764E-03 2.777OJS-03 1 e7581ET-03 1.0863JS-03 6.5584E-64 3.8728E-04 2.2392E-04 1.2689E-04 7.0548E-05 3.8519E-05 2.06753-05 1.092OE-05 5.6808E46 2.9138E-06 1*47503z-O6 7.3758E-07 3.6474E-07 1.7853E-07 8.659OE-08 4.1655E-08

T = 2000°K

at the temperatures

T = 15OO’K ---

Line intensities (in atm-lcm-2)

POSITIONS,MATRIXELEMENTS, AND LINE INTENSITIES OF LINESIN THE R-BRANCH OF THE VIBRATIONALTRANSITIONI’ = 2. TO Y = 2 IN HF-19 --

3*5672E-04 2.7607E-03 8.8122E03 1e9322E-02 3.4146E-02 5.2225E-02 7*1816E-02 9.0843E-02 l.O728E-0 1 1.1948E-01 1*2644E-01 1.2782E-01 1,2399E-O1 1.1578E-01 l.O439E-01 9.1076JS-02 7.7047E-02 6.331OE-02 5Q609E-02 3.9411E-02 2.9936E-02 2.2206E-02 1~6104E-02 1.1429E-02 7*9453E-03 5.4161E-03 3*6234E-03 2.3812E-03 1.5385E-03 9.7809E-04 6.1237E-04 3.7789E-04 2.3003Ea4 1.3823E-04 8.2077E-05 4.8189E-05 2.8OOOE-05 1.6113E-05 9.1915E-06 5.2013E-06 2.9223E-06

T = 2500°K

Values of the positional spectroscopic constants were taken from the work of D. E. MANN, B. A. THRUSH,D. R. LIDE, JR., J. J. BALL and N. ACQWTA, J. Chem. Phys. 34, 420 (1961). Values of the dipole moment expansion coefficients were taken from the work of G. A. KUIPERS,J. Molec. Spectroscopy 2. 75 (1958).

38.061 76.075 113.992 151.764 189,346 226.688 263.144 300.470 336.818 372.745 408.208 443.162 477.569 511.385 544.573 577095 608.911 639.989 670.293 699.788 728445 756.231 783.117 809.074 834.074 858.090 881.096 903 ~066 923.976 943.799 962.512 980.089 996.504 1011.732 1025.746 1038.515 1050.012 1060.202 1069.054 1076.527 1082.581

Position (in l/cm)

TABLE A-4.

--

B L

.? Q :!

ol

c! sr